Let R be the relation on the set of ordered pairs of positive integers such that ((a, b), (c, d)) ∈ R if and only if ad = bc. Determine whether R is an equivalence relation or a partial ordering.
Ans: R is described on the set P of cross product of set of positive integers Z+ as (a, b) R (c, d) iff a*d =b*c. Here now let us test if R is an equivalence relation or not
Reflexivity: Let (x, x) be any element of P, after that since a*a = a*a , we can say the (a, a) R (a, a).So R is reflexive.
Symmetry: Let (a, b) and (c, d) are any two elements in P like that (a, b) R (c, d). After that we have a*d = b*c => c*b = d*a => (c, d) R (a, b) => R is symmetric.
Transitivity: Let assume (a, b), (c, d) and (e, f) are any three pairs in P like that (a, b) R (c, d) and (c, d) R (e, f). After that we have a*d = b*c and c*f = d*e => a/e = b/f => a*f = b*e => (a, b) R (e, f) => R is transitive.
Hence R is an equivalence relation.